Conley index orientations

Abstract

The homotopy Conley index along heteroclinic solutions of certain parabolic evolution equations is zero under appropriate assumptions. This result implies that the so-called connecting homomorphism associated with a heteroclinic solution is an isomorphism. Hence, using $\mathbb{Z}$-coefficients it can be viewed as either $1$ or $-1$ - depending on the choice of generators for the homology Conley index. We develop a method to choose such generators, and compute the connecting homomorphism relative to these generators.